Problem: $\dfrac{ -6x - 2y }{ 8 } = \dfrac{ 9x - 4z }{ 4 }$ Solve for $x$.
Multiply both sides by the left denominator. $\dfrac{ -6x - 2y }{ {8} } = \dfrac{ 9x - 4z }{ 4 }$ ${8} \cdot \dfrac{ -6x - 2y }{ {8} } = {8} \cdot \dfrac{ 9x - 4z }{ 4 }$ $-6x - 2y = {8} \cdot \dfrac { 9x - 4z }{ 4 }$ Reduce the right side. $-6x - 2y = {8} \cdot \dfrac{ 9x - 4z }{ {4} }$ $-6x - 2y = {2} \cdot \left( 9x - 4z \right)$ Distribute the right side $-6x - 2y = {2} \cdot \left( {9x} - {4z} \right)$ $-6x - 2y = {18}x - {8}z$ Combine $x$ terms on the left. $-{6x} - 2y = {18x} - 8z$ $-{24x} - 2y = -8z$ Move the $y$ term to the right. $-24x - {2y} = -8z$ $-24x = -8z + {2y}$ Isolate $x$ by dividing both sides by its coefficient. $-{24}x = -8z + 2y$ $x = \dfrac{ -8z + 2y }{ -{24} }$ All of these terms are divisible by $2$ Divide by the common factor and swap signs so the denominator isn't negative. $x = \dfrac{ {4}z - {1}y }{ {12} }$